In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density …

This paper deals mainly with various ‘distribution functions’ and ‘cumulative distribution functions’ pertaining to the modulus and to the angle of the ‘normal’ complex variate, for the case where …

Characteristic functions are essentially Fourier transformations of distribution functions, which provide a general and powerful tool to analyze probability distributions. Recall that in order to …

Mar 25, 2025 · 3. Cumulative Distribution Function (CDF): it indicates the cumulated probability up to a certain value, for example, the probability of a newborn baby weighing up to 2.5kg. It is …

Aug 26, 2015 · Let $X$ be a random variable with characteristic function $\phi(\ )$ satisfying $|\phi(t)|=1$ for all $|t|\leq 1/T$ with some $T>0$. Show that $X$ is degenerate, i.e., there is …

Dec 8, 2017 · There is a standard result in probability theory that whenever $X$ and $Y$ are independent, then so are $f(X)$ and $g(Y)$ for (measurable) functions $f$ and $g$. This …

distribution of X is usually represented by, X ∼ N(µ,σ2), or also, X ∼ N(x−µ,σ2). The Normal or Gaussian pdf (1.1) is a bell-shaped curve that is symmetric about the mean µ and that attains …

Jul 1, 2017 · This paper deals mainly with various ‘distribution functions’ and ‘cumulative distribution functions’ pertaining to the modulus and to the angle of the ‘normal’ complex …

Apr 18, 2019 · Given that $X$ is uniformly distributed over $K$ (that is, the probability density function of $X$ is $f(x) = \frac{1}{2^{64}}$), what is the distribution of $Y = X \bmod k$? My gut …

Dec 8, 2013 · The characteristic function of a probability measure m on B(R) is the function jm: R!C given by jm(t) = Z eitx m(dx) When we speak of the characteristic function jX of a random …